Normalization, Cut-Elimination, and the Theory of Proofs

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Normalization, cut-elimination, and the theory of proofs

More complicated examples, with duplication of nested boxes. Definition of switchings and statement of the Sequentialisation Theorem very brief statement in [BM09] Proposition 2, details from [G96], examples from [J91] Section 6. If there is any time remaining, some details of the proof of the Sequentialisation Theorem from [G96]. A rough plan for the second seminar, which will be taken almost entirely from [BM09].

A brief recall of the relation between stratification and complexity from my earlier talk slides and screencast. A brief recall of proof-nets and their cut-elimination steps from last time. The definition of stratified proof-nets mL3 in Baillot-Mazza from Section 2. A warning: Girard's style is a little slippery, and it is common for students to say they have read it, who turn out to have absorbed the opinions but little of the results.

Normalization, Cut-Elimination, and the Theory of Proofs, Ungar

Postscript — If you care about the fine technicalities of matching up normal proofs in natural deductions with cut-free proofs in sequent calculus, Ungar, Normalization, cut-elimination, and the theory of proofs is a good text, generously made freely available as part of the Stanford Medieval and Modern Thought Digitization Project. This literature is a bit tricky, because the two proof calculi are formulated, and their metatheory have come about in a somewhat different manner. The literature doesn't date back to Gentzen, except to the trivial extent that the two calculi are shown to have equivalent expressive strength, because the theory of normalisation for natural deduction was not fixed until Prawitz, , Natural Deduction.

This chapter builds towards an explanation of sequent calculus by way of its relationship to natural deduction.

In the first lecture she tries to obtain sequent calculus from natural deduction very naturally of course, by making hypotheses explicit. If you are looking for a source on basic sequent calculus, both propositional and first-order logicby which I mean the basic definitions, explanations, and basic theorems like cut-elimination , and all this developed from scratch and in a very readable way, with all notations explainedI would recommend the book of Cook and Nguyen, Logical Foundations of Proof Complexity , Cambridge University Press, ASL Perspectives in Logic.

An online draft which is almost identical to the book is here.

I second here the recommendations of Troelstra and Schwichtenberg's Basic Proof Theory for learning sequent calculus because of its exercises; focus on chapter 3 and do the exercises to get a good grip on standard sequent calculi for classical and intuitionistic logic, as well as get some exposure to a few variations e. Sign up to join this community.

The best answers are voted up and rise to the top. Home Questions Tags Users Unanswered. Do you know any good introductory resource on sequent calculus? Ask Question. Asked 9 years, 7 months ago.

Cut-elimination theorem

Active 2 years, 5 months ago. A warning: Girard's style is a little slippery, and it is common for students to say they have read it, who turn out to have absorbed the opinions but little of the results. Postscript — If you care about the fine technicalities of matching up normal proofs in natural deductions with cut-free proofs in sequent calculus, Ungar, Normalization, cut-elimination, and the theory of proofs is a good text, generously made freely available as part of the Stanford Medieval and Modern Thought Digitization Project. This literature is a bit tricky, because the two proof calculi are formulated, and their metatheory have come about in a somewhat different manner.

The literature doesn't date back to Gentzen, except to the trivial extent that the two calculi are shown to have equivalent expressive strength, because the theory of normalisation for natural deduction was not fixed until Prawitz, , Natural Deduction. This chapter builds towards an explanation of sequent calculus by way of its relationship to natural deduction.

In the first lecture she tries to obtain sequent calculus from natural deduction very naturally of course, by making hypotheses explicit. If you are looking for a source on basic sequent calculus, both propositional and first-order logicby which I mean the basic definitions, explanations, and basic theorems like cut-elimination , and all this developed from scratch and in a very readable way, with all notations explainedI would recommend the book of Cook and Nguyen, Logical Foundations of Proof Complexity , Cambridge University Press, ASL Perspectives in Logic.

An online draft which is almost identical to the book is here. I second here the recommendations of Troelstra and Schwichtenberg's Basic Proof Theory for learning sequent calculus because of its exercises; focus on chapter 3 and do the exercises to get a good grip on standard sequent calculi for classical and intuitionistic logic, as well as get some exposure to a few variations e. Sign up to join this community. The best answers are voted up and rise to the top. Home Questions Tags Users Unanswered. Do you know any good introductory resource on sequent calculus?